what is the dimension of a vector spaceSolutionDimension is

what is the dimension of a vector space?

Solution

Dimension is the number of elements/ vectors in the basis of a vector space.

It is also the size of maximal set of the linearly independent vectors in a vector space.

For example:

C={(x,y,z) : x+y=z & x,y,z are reals}

Clearly, C is a vector space over real field.

A basis is : { (1,0,1), (0,1,1) }

So, the diemnsion is 2 becuse the basis has 2 vectors.

Note that (1,0,1) and (0,1,1) are linearly independent. But there cannot be more than 3 vectors (mutually) linearly independent.That is, there is no other vector in C that is linearly independent with the above two vectors. This is shown below-

Take any (a,b,c) belonging to C then, c=a+b

(a,b,c)= (a,b,a+b) = a*(1,0,1)+b*(0,1,1)

Hence, there is no other vector in C that is linearly independent with (1,0,1) and (0,1,0)